TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Comparison, absolute-convergence, and limit-comparison tests for noncompact Henstock–Kurzweil integrals

Statement

Let $f, g$ be HK integrable on every compact truncation near the same missing endpoint.

If $g \geq 0$ , $∣ f ∣ \leq g$ eventually, and the noncompact integral of $g$ converges, then that of $f$ converges.
If the noncompact integral of $∣ f ∣$ converges, then that of $f$ converges.
If $f, g > 0$ eventually and $f / g \to c$ with $0 < c < \infty$ , their noncompact integrals converge or diverge together. If $c = 0$ , convergence for $g$ implies convergence for $f$ ; if $c = + \infty$ , convergence for $f$ implies convergence for $g$ .

The corresponding assertions hold at finite and infinite missing endpoints on either side. At an infinite missing endpoint, the notation $f / g \to + \infty$ in claim 3 means explicitly that for every real $M > 0$ , one has $f / g > M$ throughout some sufficiently late tail; this clause does not rely on a finite-limit definition.

Facts & Assumptions

Given: The locally integrable functions and eventual inequalities in the Statement.

[L1]

If $p$ and $q$ are HK integrable on a compact interval and $p \leq q$ there, then $\int p \leq \int q$ (Monotonicity of the Henstock–Kurzweil integral).

[L2]

Noncompact integrability is equivalent to uniformly small tail integrals (The Cauchy criterion for a Henstock–Kurzweil integral at a missing endpoint).

[L3]

If $p$ and $q$ are HK integrable on a compact interval, then every linear combination is HK integrable and its integral is the same linear combination of their integrals (Linearity of the Henstock–Kurzweil integral).

Proof

technique · direct

1.1

On every sufficiently late compact tail, $- g \leq f \leq g$ . By [L3], $- g$ is integrable with integral $- \int g$ , so two applications of [L1] give $- \int g \leq \int f \leq \int g$ and hence $∣ \int f ∣ \leq \int g$ ; the tail criterion [L2] proves claim 1, and taking $g = ∣ f ∣$ proves claim 2.

givenL1L2L3

2.1

If $f / g \to c \in (0, \infty)$ , it lies between two positive constants near the endpoint, so two applications of step 1.1 give equivalence; for limit $0$ or $+ \infty$ , the corresponding one-sided eventual bound gives exactly the stated implication.

step 1.1algebra∎

Depends on

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Sources

Alessandro Fonda, The Kurzweil-Henstock Integral for Undergraduates, Ch. 1 (standard reference, not scraped)
Andrew Bruckner, Judith Bruckner and Brian Thomson, Real Analysis, Section 1.21 (standard reference, not scraped)